\(\int x^{-1-n (3+p)} (a+b x^n)^p (c+d x^n)^2 \, dx\) [514]

Optimal result

Integrand size = 29, antiderivative size = 160 \[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=-\frac {\left (2 b^2 c^2-2 a b c d (3+p)+a^2 d^2 \left (6+5 p+p^2\right )\right ) x^{-n (1+p)} \left (a+b x^n\right )^{1+p}}{a^3 n (1+p) (2+p) (3+p)}+\frac {2 c (b c-a d (3+p)) x^{-n (2+p)} \left (a+b x^n\right )^{1+p}}{a^2 n (2+p) (3+p)}-\frac {c^2 x^{-n (3+p)} \left (a+b x^n\right )^{1+p}}{a n (3+p)} \] Output:

-(2*b^2*c^2-2*a*b*c*d*(3+p)+a^2*d^2*(p^2+5*p+6))*(a+b*x^n)^(p+1)/a^3/n/(p+ 
1)/(2+p)/(3+p)/(x^(n*(p+1)))+2*c*(b*c-a*d*(3+p))*(a+b*x^n)^(p+1)/a^2/n/(2+ 
p)/(3+p)/(x^(n*(2+p)))-c^2*(a+b*x^n)^(p+1)/a/n/(3+p)/(x^(n*(3+p)))
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.64 \[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=-\frac {c^2 x^{-n (3+p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (1+\frac {d x^n}{c}\right )^{3+p} \operatorname {Hypergeometric2F1}\left (-3-p,-p,-2-p,\frac {-\frac {b x^n}{a}+\frac {d x^n}{c}}{1+\frac {d x^n}{c}}\right )}{n (3+p)} \] Input:

Integrate[x^(-1 - n*(3 + p))*(a + b*x^n)^p*(c + d*x^n)^2,x]
 

Output:

-((c^2*(a + b*x^n)^p*(1 + (d*x^n)/c)^(3 + p)*Hypergeometric2F1[-3 - p, -p, 
 -2 - p, (-((b*x^n)/a) + (d*x^n)/c)/(1 + (d*x^n)/c)])/(n*(3 + p)*x^(n*(3 + 
 p))*(1 + (b*x^n)/a)^p))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.45, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1008, 25, 959, 803, 803, 796}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^{-n (p+3)-1} \left (c+d x^n\right )^2 \left (a+b x^n\right )^p \, dx\)

\(\Big \downarrow \) 1008

\(\displaystyle -\frac {\int -x^{-n (p+3)-1} \left (b x^n+a\right )^p \left (c n (b c-a d (p+3))-a d^2 n (p+2) x^n\right )dx}{b n}-\frac {d x^{-n (p+3)} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int x^{-n (p+3)-1} \left (b x^n+a\right )^p \left (c n (b c-a d (p+3))-a d^2 n (p+2) x^n\right )dx}{b n}-\frac {d x^{-n (p+3)} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b n}\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {\frac {n \left (a^2 d^2 \left (p^2+5 p+6\right )-2 a b c d (p+3)+2 b^2 c^2\right ) \int x^{-n (p+3)-1} \left (b x^n+a\right )^pdx}{2 b}+\frac {a d^2 (p+2) x^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{2 b}}{b n}-\frac {d x^{-n (p+3)} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b n}\)

\(\Big \downarrow \) 803

\(\displaystyle \frac {\frac {n \left (a^2 d^2 \left (p^2+5 p+6\right )-2 a b c d (p+3)+2 b^2 c^2\right ) \left (-\frac {2 b \int x^{-n (p+2)-1} \left (b x^n+a\right )^pdx}{a (p+3)}-\frac {x^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{a n (p+3)}\right )}{2 b}+\frac {a d^2 (p+2) x^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{2 b}}{b n}-\frac {d x^{-n (p+3)} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b n}\)

\(\Big \downarrow \) 803

\(\displaystyle \frac {\frac {n \left (a^2 d^2 \left (p^2+5 p+6\right )-2 a b c d (p+3)+2 b^2 c^2\right ) \left (-\frac {2 b \left (-\frac {b \int x^{-n (p+1)-1} \left (b x^n+a\right )^pdx}{a (p+2)}-\frac {x^{-n (p+2)} \left (a+b x^n\right )^{p+1}}{a n (p+2)}\right )}{a (p+3)}-\frac {x^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{a n (p+3)}\right )}{2 b}+\frac {a d^2 (p+2) x^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{2 b}}{b n}-\frac {d x^{-n (p+3)} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b n}\)

\(\Big \downarrow \) 796

\(\displaystyle \frac {\frac {n \left (a^2 d^2 \left (p^2+5 p+6\right )-2 a b c d (p+3)+2 b^2 c^2\right ) \left (-\frac {2 b \left (\frac {b x^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a^2 n (p+1) (p+2)}-\frac {x^{-n (p+2)} \left (a+b x^n\right )^{p+1}}{a n (p+2)}\right )}{a (p+3)}-\frac {x^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{a n (p+3)}\right )}{2 b}+\frac {a d^2 (p+2) x^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{2 b}}{b n}-\frac {d x^{-n (p+3)} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b n}\)

Input:

Int[x^(-1 - n*(3 + p))*(a + b*x^n)^p*(c + d*x^n)^2,x]
 

Output:

-((d*(a + b*x^n)^(1 + p)*(c + d*x^n))/(b*n*x^(n*(3 + p)))) + ((a*d^2*(2 + 
p)*(a + b*x^n)^(1 + p))/(2*b*x^(n*(3 + p))) + (n*(2*b^2*c^2 - 2*a*b*c*d*(3 
 + p) + a^2*d^2*(6 + 5*p + p^2))*(-((a + b*x^n)^(1 + p)/(a*n*(3 + p)*x^(n* 
(3 + p)))) - (2*b*((b*(a + b*x^n)^(1 + p))/(a^2*n*(1 + p)*(2 + p)*x^(n*(1 
+ p))) - (a + b*x^n)^(1 + p)/(a*n*(2 + p)*x^(n*(2 + p)))))/(a*(3 + p))))/( 
2*b))/(b*n)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 796

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, 
 p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
 

rule 803

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( 
a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 
)))   Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I 
LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
 

rule 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 1008

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n) 
^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Simp[1/(b*(m + n*(p + q) + 1)) 
Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + 
 c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d* 
n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c 
 - a*d, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(729\) vs. \(2(163)=326\).

Time = 8.20 (sec) , antiderivative size = 730, normalized size of antiderivative = 4.56

method	result	size
parallelrisch	\(-\frac {x \,x^{3 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{2} b^{2} d^{2} p^{2}+5 x \,x^{3 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{2} b^{2} d^{2} p -2 x \,x^{3 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a \,b^{3} c d p +x \,x^{2 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{3} b \,d^{2} p^{2}+2 x \,x^{2 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{2} b^{2} c d \,p^{2}+6 x \,x^{3 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{2} b^{2} d^{2}-6 x \,x^{3 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a \,b^{3} c d +2 x \,x^{3 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} b^{4} c^{2}+5 x \,x^{2 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{3} b \,d^{2} p +6 x \,x^{2 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{2} b^{2} c d p -2 x \,x^{2 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a \,b^{3} c^{2} p +2 x \,x^{n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{3} b c d \,p^{2}+x \,x^{n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{2} b^{2} c^{2} p^{2}+6 x \,x^{2 n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{3} b \,d^{2}+8 x \,x^{n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{3} b c d p +x \,x^{n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{2} b^{2} c^{2} p +x \,x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{3} b \,c^{2} p^{2}+6 x \,x^{n} x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{3} b c d +3 x \,x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{3} b \,c^{2} p +2 x \,x^{-n p -3 n -1} \left (a +b \,x^{n}\right )^{p} a^{3} b \,c^{2}}{\left (p^{2}+5 p +6\right ) n \left (p +1\right ) a^{3} b}\)	\(730\)

Input:

int(x^(-1-n*(3+p))*(a+b*x^n)^p*(c+d*x^n)^2,x,method=_RETURNVERBOSE)
 

Output:

-(x*(x^n)^3*x^(-n*p-3*n-1)*(a+b*x^n)^p*a^2*b^2*d^2*p^2+5*x*(x^n)^3*x^(-n*p 
-3*n-1)*(a+b*x^n)^p*a^2*b^2*d^2*p-2*x*(x^n)^3*x^(-n*p-3*n-1)*(a+b*x^n)^p*a 
*b^3*c*d*p+x*(x^n)^2*x^(-n*p-3*n-1)*(a+b*x^n)^p*a^3*b*d^2*p^2+2*x*(x^n)^2* 
x^(-n*p-3*n-1)*(a+b*x^n)^p*a^2*b^2*c*d*p^2+6*x*(x^n)^3*x^(-n*p-3*n-1)*(a+b 
*x^n)^p*a^2*b^2*d^2-6*x*(x^n)^3*x^(-n*p-3*n-1)*(a+b*x^n)^p*a*b^3*c*d+2*x*( 
x^n)^3*x^(-n*p-3*n-1)*(a+b*x^n)^p*b^4*c^2+5*x*(x^n)^2*x^(-n*p-3*n-1)*(a+b* 
x^n)^p*a^3*b*d^2*p+6*x*(x^n)^2*x^(-n*p-3*n-1)*(a+b*x^n)^p*a^2*b^2*c*d*p-2* 
x*(x^n)^2*x^(-n*p-3*n-1)*(a+b*x^n)^p*a*b^3*c^2*p+2*x*x^n*x^(-n*p-3*n-1)*(a 
+b*x^n)^p*a^3*b*c*d*p^2+x*x^n*x^(-n*p-3*n-1)*(a+b*x^n)^p*a^2*b^2*c^2*p^2+6 
*x*(x^n)^2*x^(-n*p-3*n-1)*(a+b*x^n)^p*a^3*b*d^2+8*x*x^n*x^(-n*p-3*n-1)*(a+ 
b*x^n)^p*a^3*b*c*d*p+x*x^n*x^(-n*p-3*n-1)*(a+b*x^n)^p*a^2*b^2*c^2*p+x*x^(- 
n*p-3*n-1)*(a+b*x^n)^p*a^3*b*c^2*p^2+6*x*x^n*x^(-n*p-3*n-1)*(a+b*x^n)^p*a^ 
3*b*c*d+3*x*x^(-n*p-3*n-1)*(a+b*x^n)^p*a^3*b*c^2*p+2*x*x^(-n*p-3*n-1)*(a+b 
*x^n)^p*a^3*b*c^2)/(p^2+5*p+6)/n/(p+1)/a^3/b
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.88 \[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=-\frac {{\left ({\left (a^{2} b d^{2} p^{2} + 2 \, b^{3} c^{2} - 6 \, a b^{2} c d + 6 \, a^{2} b d^{2} - {\left (2 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} p\right )} x x^{-n p - 3 \, n - 1} x^{3 \, n} + {\left (6 \, a^{3} d^{2} + {\left (2 \, a^{2} b c d + a^{3} d^{2}\right )} p^{2} - {\left (2 \, a b^{2} c^{2} - 6 \, a^{2} b c d - 5 \, a^{3} d^{2}\right )} p\right )} x x^{-n p - 3 \, n - 1} x^{2 \, n} + {\left (6 \, a^{3} c d + {\left (a^{2} b c^{2} + 2 \, a^{3} c d\right )} p^{2} + {\left (a^{2} b c^{2} + 8 \, a^{3} c d\right )} p\right )} x x^{-n p - 3 \, n - 1} x^{n} + {\left (a^{3} c^{2} p^{2} + 3 \, a^{3} c^{2} p + 2 \, a^{3} c^{2}\right )} x x^{-n p - 3 \, n - 1}\right )} {\left (b x^{n} + a\right )}^{p}}{a^{3} n p^{3} + 6 \, a^{3} n p^{2} + 11 \, a^{3} n p + 6 \, a^{3} n} \] Input:

integrate(x^(-1-n*(3+p))*(a+b*x^n)^p*(c+d*x^n)^2,x, algorithm="fricas")
 

Output:

-((a^2*b*d^2*p^2 + 2*b^3*c^2 - 6*a*b^2*c*d + 6*a^2*b*d^2 - (2*a*b^2*c*d - 
5*a^2*b*d^2)*p)*x*x^(-n*p - 3*n - 1)*x^(3*n) + (6*a^3*d^2 + (2*a^2*b*c*d + 
 a^3*d^2)*p^2 - (2*a*b^2*c^2 - 6*a^2*b*c*d - 5*a^3*d^2)*p)*x*x^(-n*p - 3*n 
 - 1)*x^(2*n) + (6*a^3*c*d + (a^2*b*c^2 + 2*a^3*c*d)*p^2 + (a^2*b*c^2 + 8* 
a^3*c*d)*p)*x*x^(-n*p - 3*n - 1)*x^n + (a^3*c^2*p^2 + 3*a^3*c^2*p + 2*a^3* 
c^2)*x*x^(-n*p - 3*n - 1))*(b*x^n + a)^p/(a^3*n*p^3 + 6*a^3*n*p^2 + 11*a^3 
*n*p + 6*a^3*n)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=\text {Timed out} \] Input:

integrate(x**(-1-n*(3+p))*(a+b*x**n)**p*(c+d*x**n)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=\int { {\left (d x^{n} + c\right )}^{2} {\left (b x^{n} + a\right )}^{p} x^{-n {\left (p + 3\right )} - 1} \,d x } \] Input:

integrate(x^(-1-n*(3+p))*(a+b*x^n)^p*(c+d*x^n)^2,x, algorithm="maxima")
 

Output:

integrate((d*x^n + c)^2*(b*x^n + a)^p*x^(-n*(p + 3) - 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^(-1-n*(3+p))*(a+b*x^n)^p*(c+d*x^n)^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{-1,[1,0,4,3,1,3,3,2,0]%%%}+%%%{-3,[1,0,4,3,1,3,2,2,0]%%%}+ 
%%%{-3,[1
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=\int \frac {{\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^2}{x^{n\,\left (p+3\right )+1}} \,d x \] Input:

int(((a + b*x^n)^p*(c + d*x^n)^2)/x^(n*(p + 3) + 1),x)
 

Output:

int(((a + b*x^n)^p*(c + d*x^n)^2)/x^(n*(p + 3) + 1), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.92 \[ \int x^{-1-n (3+p)} \left (a+b x^n\right )^p \left (c+d x^n\right )^2 \, dx=\frac {\left (x^{n} b +a \right )^{p} \left (-x^{3 n} a^{2} b \,d^{2} p^{2}-5 x^{3 n} a^{2} b \,d^{2} p -6 x^{3 n} a^{2} b \,d^{2}+2 x^{3 n} a \,b^{2} c d p +6 x^{3 n} a \,b^{2} c d -2 x^{3 n} b^{3} c^{2}-x^{2 n} a^{3} d^{2} p^{2}-5 x^{2 n} a^{3} d^{2} p -6 x^{2 n} a^{3} d^{2}-2 x^{2 n} a^{2} b c d \,p^{2}-6 x^{2 n} a^{2} b c d p +2 x^{2 n} a \,b^{2} c^{2} p -2 x^{n} a^{3} c d \,p^{2}-8 x^{n} a^{3} c d p -6 x^{n} a^{3} c d -x^{n} a^{2} b \,c^{2} p^{2}-x^{n} a^{2} b \,c^{2} p -a^{3} c^{2} p^{2}-3 a^{3} c^{2} p -2 a^{3} c^{2}\right )}{x^{n p +3 n} a^{3} n \left (p^{3}+6 p^{2}+11 p +6\right )} \] Input:

int(x^(-1-n*(3+p))*(a+b*x^n)^p*(c+d*x^n)^2,x)
 

Output:

((x**n*b + a)**p*( - x**(3*n)*a**2*b*d**2*p**2 - 5*x**(3*n)*a**2*b*d**2*p 
- 6*x**(3*n)*a**2*b*d**2 + 2*x**(3*n)*a*b**2*c*d*p + 6*x**(3*n)*a*b**2*c*d 
 - 2*x**(3*n)*b**3*c**2 - x**(2*n)*a**3*d**2*p**2 - 5*x**(2*n)*a**3*d**2*p 
 - 6*x**(2*n)*a**3*d**2 - 2*x**(2*n)*a**2*b*c*d*p**2 - 6*x**(2*n)*a**2*b*c 
*d*p + 2*x**(2*n)*a*b**2*c**2*p - 2*x**n*a**3*c*d*p**2 - 8*x**n*a**3*c*d*p 
 - 6*x**n*a**3*c*d - x**n*a**2*b*c**2*p**2 - x**n*a**2*b*c**2*p - a**3*c** 
2*p**2 - 3*a**3*c**2*p - 2*a**3*c**2))/(x**(n*p + 3*n)*a**3*n*(p**3 + 6*p* 
*2 + 11*p + 6))